initial error vectors, then the matrix H will map them to a set of final orthogonal error vectors of length . The action of the system maps an infinitesimal sphere of inititial points to an ellipsoid whose major axis is given by the while the minor axis is given by , where N is the number of dimensions. This definition of Lyapunov exponents is both more elegant and more appropriate to real-world, continuous-time dynamical systems than the more usual definition based on discrete function maps. Chaos is defined as the existence of at least one positive Lyapunov exponent. In a chaotic system, we call the Lyapunov exponent the asymptotic value of the greatest eigenvalue of H: If there is any significant difference between the Lyapunov exponents then as an error vector evolves forward in time, any displacement in the direction of largest growth will tend to be magnified. Thus: The Lyapunov exponent of a flow is a unique quantity, that characterizes the asymptotic separation of fluid particles in a given flow. It is often used as a measure of the efficiency of mixing, since it measures how fast trajectories separate from each other because of chaotic advection. The Lyapunov exponent can be computed by different methods: by following one single trajectory for very long times and computing . or by following an ensemble of trajectories for a given period of time, and computing the ensemble average: The equivalence of the two methods is due to the ergodicity of the chaotic system. Filament growth versus evolution of the tracer gradient The following, exact equation can be derived from an advection-diffusion equation (see below), with a diffusion term (D=0) of zero: In parallel with the definition of the Lyapunov exponent, we define the matrix , as follows: It is easy to show that: If we define as the squared lengths of the principal components of the tracer gradient matrix, , then: where the 's are arranged, as before, from largest to smallest. Therefore, growth in the error vector will cause a corresponding decrease in the tracer gradient and vice versa. This can be understood very simply and intuitively by considering two nearby points: since the difference in tracer concentration will be fixed, the only source of variation in the gradients between them will be their separation. Contour advection Contour advection is another useful method for characterizing chaotic mixing. In chaotic flows, advected